Welcome to the ReMInDeRY!
(Research Methods: Interactive Demonstrations in R at York University)
Here we host a set of interactive demonstrations for concepts related to research methods. These demonstrations will also introduce you to the statistical program called R.
If you are just starting the tutorial, please begin by clicking ‘Start Over’ at the bottom of the index to your left.
If you have any questions and comments, please contact Dr. Raymond Mar (mar@yorku.ca)
UNIT 3: Measures of Central Tendency (Part 2) and Variability
GOALS:
- Explore how the spread of scores influences different measures of central tendency.
- Visualize your data and examine its distribution by producing a histogram.
- Manipulate data by adding and combining it, or generating it.
In Unit 2, we learned about 3 measures of central tendency: the mean, the median, and the mode. All 3 help us to summarize a lot of data, making all these numbers easier to understand and to work with. However, the 3 measures of central tendency are also different, meaning different things and serving different purposes.
- The Mean is the average response across all of the data. It is the most commonly used and is typically what people mean when they talk about the average of something.
- The Median is the middle response, the response right in the middle of the distribution of the data. You could think of this as the average person’s individual response, perhaps. It is less commonly reported and used, but is very useful in some instances and superior to the Mean in some cases.
- The Mode is the most frequent or most common response. It is the answer to the question, “What do most people say?”
Differences between the Mean and the Median
Sometimes the Mean and Median are very similar. Let’s take a look at
our data from the last unit, which reported all the Cats and Dogs
registered within each postal code area of Toronto. I’ve taken the
liberty of loading the data into an object called
TOcatsdogs, just like in Unit 2. This object,
TOcatsdogs now contains our data on pet ownership in
Toronto.
Use the R window below to find the Mean and the Median for the number of Dogs per postal code area in Toronto, then use your answers to complete the Quiz below. You may want to first familiarize yourself with what these data look like. Do you remember how we view the first few rows of an object from Unit 2? If you don’t remember, you can always review Unit 2, or search for the answer online.
A document that contains lines of code is known as a SCRIPT, and a script will typically contain many different functions with each function on a different line. So you can write a small script in the box below to look at the first few lines of the data, then find the Mean and Median for the number of Dogs per postal code in Toronto.
Hint: You should know the functions to find the
mean() and the median(), now you just need to
put the right data-object within the parentheses. Remember that the
whole dataset is called TOcatsdogs and that we use a
$ to specify a particular column or variable of interest,
which in this case is DOG. So the object we’re interested
in is TOcatsdogs$DOG.
Differences between the Mean and the Median
Excellent work! You can see that the Mean number of dogs and the Median number of dogs is actually rather similar: 562.5 versus 517.
So you might be wondering why we need the Median at all. Well, the thing about the Mean is that it averages across all of the data and therefore it can be highly affected by Outliers. Outliers are values or numbers that are rather extreme, so much smaller or larger than all the rest of the numbers.
For example, let’s take our data of Cats and Dogs and add a new row that looks like this:
| AREA | CAT | DOG | TOTAL |
|---|---|---|---|
| M9Z | 1 | 6999 | 7000 |
In other words, we’re adding an imaginary new postal code (M9Z) in Toronto in which there are registered 6,999 Dogs and 1 lonely Cat.
To add these data to our dataset (located in our object,
TOcatsdogs), we’d use the following line of code:
NewRow <- data.frame(AREA="M9Z", CAT = 1, DOG = 6999, TOTAL = 7000)
TOcatsdogs_outlier <- rbind (TOcatsdogs, NewRow)In this code, we first make a new dataset that includes the
information we want to add to TOcatsdogs, and we’re going
to place it in an object that we call NewRow. To do this,
we’re using the data.frame() function to create a new
data-frame, that includes the information we’re adding, with each value
associated with a column name or header (these match the column names
for our existing dataset: AREA, CAT, DOG, TOTAL). Next, we’re going to
use the rbind() command, which stands for Row Bind, or bind
two objects together by their rows (as opposed to their columns, which
would be cbind()). The two objects we’re joining together
are our old dataset, located in the object TOcatsdogs, and
our new one, in the object NewRow. Once we’ve joined these
two things together, we’ll place them into a new object, which we’re
calling TOcatsdogs_outlier. Remember that we typically put
the result of any operation or function into an object, that way we can
manipulate the result further. Because we are often putting datasets
into objects, we can often use these terms interchangeably. Make
sense?
Let’s take a look to see if everything worked properly. In order to
do so, we’re going to learn a new function: tail(). This
function is just like head(), but instead of showing us the
first few rows of a dataset, it shows us the last few. This way we can
look and see if we were successful in adding this new row of data to our
dataset. Give it a try below!
NewRow <- data.frame(AREA="M9Z", CAT = 1, DOG = 6999, TOTAL = 7000)
TOcatsdogs_outlier <- rbind (TOcatsdogs, NewRow)Hint: Add the code to look at the end of the dataset
to what’s already in the box, then run everything. Be sure to put the
name of the new dataset (with the outlier) into the
parentheses for tail().
Differences between the Mean and the Median (2)
If everything worked properly, you should see a 99th row, for postal
code M9Z at the end of our TOcatsdogs_outlier dataset.
Ok, let’s compare the mean and the median, now that we’ve added this
outlier. But let’s also learn a new function that can help us do so. The
package called psych includes all kinds of useful
functions. How would we go about installing and loading this
package?
Differences between the Mean and the Median (3)
Great! Now that this package has been loaded we can use one of its
very useful functions, describe(). The
describe() function provides a lot of useful descriptives
for a variable, including the mean and the median. Use what you know
about R to find the mean and the median using the
describe() function for our new dataset
(TOcatsdogs_outlier) and the variable we’re interested in,
DOG.
describe()describe(TOcatsdogs_outlier$D__)describe(TOcatsdogs_outlier$DOG)If the whole output doesn’t fit on your screen, you’ll find a small black arrow at the right of the table, clicking that will reveal the rest of the table (clicking the arrow that appears on the left allows you to go back).
Differences between the Mean and the Median (3)
Now you can see that our mean and our median differ quite a lot for this variable! By just adding one row of data, a new postal code that has 6,999 dogs in it, the mean becomes 627.5 whereas the median becomes 521. Without this one new piece of data–which is an outlier (or extreme value, much larger than the others)–the mean (562.5) and the median (517) were relatively similar. But now, with the outlier, the mean changes a lot (by about 65 dogs) whereas the median changes very little (by 3 dogs).
If you think about why this happens, it makes a lot of sense. To find the median, you simply find the data-point that is right in the middle of the distribution after you’ve sorted the data from smallest to biggest. So adding our one outlier (6,999) means shifting the median over just a tiny bit (by one value). In contrast, to calculate the mean, we add all the numbers together and then divide by how many numbers we have. This means adding 6,999 to our previous sum, which is a lot! So even after we divide by our number of postal codes, it still means a big increase for the mean.
Because means can be more highly influenced by outliers than medians, means can provide a poor picture of the overall data when an outlier (or outliers) exists. This is why it’s often very useful to look at medians as well as means.
Variability
Summary statistics, like our measures of central tendency (i.e., the mean, median, and mode), attempt to summarize a large amount of data. But it’s hard to use just one number to properly summarize or represent a whole bunch of data (i.e., a whole lot of numbers).
Let’s take a look at our data on cats and dogs in Toronto using a
histogram or frequency chart. This is a basic figure showing possible
values (or responses) along the bottom (the x-axis) and then how many
data-points there are for these values (along the upright y-axis). In
other words, for our data, this is a very simple visualization of how
many postal codes have about 100 pets registered in them, or 400, or
1300. Similar to how we used head() and tail()
to look at our data, creating a histogram using the hist()
function is just another way of looking at or visualizing our data. Try
using the hist() function to produce a histogram, for the
TOTAL number of cats and dogs per postal code in
Toronto!
hist()hist(TOcatsdogs)hist(TOcatsdogs$T____)hist(TOcatsdogs$TOTAL)Distributions of Data an Variability
Looking at our histogram, it seems like most of our data is clumped somewhere between 500 and 1000 pets per postal code, so around 750 would be a good guess for our measures of central tendency like the mean and the median. You can see this in the histogram because the highest bars (indicating the most frequent scores) like within this range.
It turns out that the mean is 814.5 and the median is 762 (you can use your R skills in the window above to double-check this), so 750 was a pretty good guess.
But you can probably also see how the mean and the median don’t actually tell us everything about our data. For example, it doesn’t tell us how spread out the data are.
As an example, consider these two histograms:
It turns out, that these two sets of data have the exact same mean (they also have the same number of data-points, in case you’re curious). So if all I told you was the mean, you might think that these datasets are identical or at least very similar. But, you can see for yourself that these two sets of data are in fact very different. The way in which they differ is in how spread out the data are, or how much each individual data-point differs from one another. This is known as VARIABILITY, or the degree to which the data-points vary or differ from one another. How many different values or responses do we have? That’s variability. If everyone gives the same answer, then we have no variability. If everyone gives a very different answers, then we have a high degree of variability.
Although there are many different measures of variability, we’re going to learn about once, know as the standard deviation.
The Standard Deviation
The standard deviation is in essence an average of how far away each individual data-point is from the mean, often abbreviated as SD. If all the data are clustered very close to the mean, then we have low variability and a smaller SD. In contrast, if all our data are very spread out and many data points are far away from the mean, we have greater variability and a larger SD.
Below is the R code that I used to generate data and produce the two histograms previously. Take a close look at this code and try to figure out how to make your own histograms with different SDs (e.g., small and large SDs). This will help you get a sense for how SD relates to the spread of data.
set.seed(1234)
x <- rnorm(n=300, mean=100, sd=15)
hist (x, xlim = c(-150, 310))x <- rnorm(n=300, mean=100, sd=15) #### This is the relevant line.x <- rnorm(n=300, mean=100, sd=15)
#### Try changing the value for SD, which is now '15', to different numbers and see what happens.#### If you like, you can fool around with changing the other parameters to see how they affect the output.The Standard Deviation (2)
Ok, now that you’ve gotten a sense of the SD, let’s take another look at our data for cats and dogs in Toronto, including the outlier (i.e., the postal code we added with 6,999 dogs in it).
Practice your R skills by using the describe() function
from the psych package to get some descriptive statistics
for the TOTAL number of pets in Toronto, using our dataset
that includes the outlier: TOcatsdogs_outlier. Then do the
same for the numbers for CAT and DOG in the
same dataset. Practice writing a longer script by placing all 3 of the
needed lines of code on a different line, to get the different results.
Then use what you’ve learned to answer the Quiz questions below.
Hint: You have already learned all you need to know
to complete this exercise. If you’re feeling stuck, go back and revisit
the previous units to refresh you memory on how to use
describe(), and how to do identify specific columns like
CAT and TOTAL in a dataset. Be sure you’re
using the right dataset, the one with the outlier already included
(i.e. ,TOcatsdogs_outlier).
Unit 3 Summary
CONGRATULATIONS! You’ve completed Unit 3 of the Remindery. In this unit you learned:
- How outliers affect the mean and median differently.
- Another form of summary statistic, related to the variability or spread of data.
- What a histogram is and how it can be used to visualize your data.
- How to create histograms using R.
- One measure of variability, known as the Standard Deviation (SD).
- How distributions of data with different SDs look.
- How outliers can affect the SD.
And hopefully you’ve become a bit more confident in your use of R to think about all these things.
If you’re interested in learning more about R, perhaps downloading it to your own computer or exploring other resources for learning R, you can find them on the ReMInDeRY website.
IMPORTANT NOTE FOR YORK STUDENTS:
PSYC 2030 (Research Methods) students: In order to get full marks for this assignment you must complete the Quiz for this tutorial, mentioned in the eClass assignment.
PSYC 2020 (Statistics) students: Click this link and enter all requested information carefully and accurately in order to receive your points for completing this tutorial.